Google Now has a neat feature (at least I think it's from Google Now) where it warns you for emergency events. The problem is it's just warned me twice in half an hour with an extremely loud tone about a snowstorm I've known about for days.

Is there some way I can make the phone vibrate or change the ringtone? I only see an option to disable these alerts entirely. It's not my default notification/alarm ringtone; it's a special LOUD warning tone. Good at attracting attention but major overkill. I would like to still get them, but I would like them to be much quieter (yes, even though they're "emergencies") as my phone blaring loud warning tones at work is quite distracting.

4 Answers
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If you go to google now, and then go to settings (the three dot menu button if its a soft key) theres a section that says notifications. That section has a part titled Urgent Updates and has the option to set a ringtone or turn on vibrate. As far as I know this should change the sound for the alert since it says urgent updates include public alerts. Just note, I am using KitKat so if you don't have the latest version of google now it could be different.
EDIT: I just realized this question was asked a year ago so the answer details a newer version

Unfortunately I don't have a working phone on hand at the moment to test this, but if you can identify the specific audio file used as the alert sound, you could simply edit it with a piece of software like Audacity and make it as quiet as you'd like. (Excuse the shoddy answer, I'll edit it as soon as I can test the method or delete it straight away if it's unsatisfactory)

I ran into the exact same issue (same storm too, no doubt). In Android 4.2, you can disable these alerts by going to Settings → Wireless Networks (More...) → Cell broadcasts. The alerts cannot be muted, but can be disabled in this screen.